Partial Identifiability for Nonnegative Matrix Factorization

Given a nonnegative matrix factorization, $R$, and a factorization rank, $r$, Exact nonnegative matrix factorization (Exact NMF) decomposes $R$ as the product of two nonnegative matrices, $C$ and $S$ with $r$ columns, such as $R = CS^\top$. A central research topic in the literature is the conditions under which such a decomposition is unique/identifiable, up to trivial ambiguities. In this paper, we focus on partial identifiability, that is, the uniqueness of a subset of columns of $C$ and $S$. We start our investigations with the data-based uniqueness (DBU) theorem from the chemometrics literature. The DBU theorem analyzes all feasible solutions of Exact NMF, and relies on sparsity conditions on $C$ and $S$. We provide a mathematically rigorous theorem of a recently published restricted version of the DBU theorem, relying only on simple sparsity and algebraic conditions: it applies to a particular solution of Exact NMF (as opposed to all feasible solutions) and allows us to guarantee the partial uniqueness of a single column of $C$ or $S$. Second, based on a geometric interpretation of the restricted DBU theorem, we obtain a new partial identifiability result. We prove it is stronger than the restricted DBU theorem, given that a proper preprocessing on the Exact NMF is used. This geometric interpretation also leads us to another partial identifiability result in the case $r=3$. Third, we show how partial identifiability results can be used sequentially to guarantee the identifiability of more columns of $C$ and $S$. We illustrate these results on several examples, including one from the chemometrics literature.